# Hypercomplex Analysis Essay

## 1. Introduction

Visual attention is an important cognitive mechanism of human survival. Humans have the capability of rapidly focusing on potential objects in a cluttered visual world based on selective visual attention, which has been studied in physiology, psychology, neural systems and computer vision for a long time [1]. The salient objects or regions often contain important semantic content, which could be applied to visual semantic analysis, such as visual servoing of autonomous mobile robots [2], motion object detection [3], image/video segmentation [4,5], scene recognition [6], smart video surveillance [7], object recognition [8] and image compression [9].

Visual saliency is a perceptual quality that makes an object distinguishable to its neighbors and, thus, captures our attention. Existing saliency approaches can be divided into two categories: task-driven attention (top-down) and data-driven attention (bottom-up). The top-down approach is a result of long-term visual simulation with prior knowledge. It is slow and task driven [10–12]. The bottom-up approach is based on low-level visual features simulating the formation of the short-term visual attention. In contrast to the top-down method, the bottom-up approach is rapid and without prior knowledge. It is a data contrast-driven mechanism in pre-attentive vision for salient objects without task [1,13–28]. In this paper, we only focus on the bottom-up approach.

Compared with task-driven visual attention, which is not clear yet, data-driven visual attention is studied extensively. Since the well-known feature integration theory (FIT) was published by A. Treisman and G. Gelade [29], there has been a growing interest in data-driven attention. Among these models, Itti and Koch's model [13] is the most famous one. They detected a saliency map by the center-surround operator and normalizing a set of low-level features. Based on the Itti's model, N. Bruce et al. proposed an information maximization detection model [14]. Liu and Zheng modeled visual attention by a CRF (conditional random field) learning algorithm [1]. Goferman introduced context information in salient object detection [25]. However, most of the methods usually need some ad hoc parameters or high-cost preprocessing, and they have difficulty in rapidly detecting a salient object.

Recently, visual saliency perception in the frequency domain has become popular. Hou [26] proposed a fast Fourier transform spectral residual analysis algorithm for image saliency detection. In this method, amplitude spectral residual is considered as an important factor to stimulate visual attention. Furthermore, Guo [27] proposed a saliency detection algorithm by using the phase spectrum of the quaternion Fourier transform. Achanta [28] gave a simple and effective salient region detection solution by the frequency-tuned method. However, for saliency perception, the problem is, which one is more important, the amplitude spectrum or the phase spectrum? Meanwhile, how does one implement visual saliency perception processing in computing parallelism? In this paper, we argue that the phase spectrum contains image structure information, and the amplitude spectrum carries the visual perception magnitude information. Based on the theories of [29,30] and the saliency detection methods of [1,13,25–28], we propose a computing parallelism algorithm named HSC, considering both amplitude spectrum and phase spectrum in a multi-scale hypercomplex of HSV (hue, saturation and value) color space and motion feature (see Section 3 for details):

In the frequency domain, amplitude spectrum and phase spectrum are both significant for saliency detection. Either one of them could not reconstruct a whole saliency map in the frequency domain.

A saliency map is the product of various visual features of comprehensive stimulation. United multi-feature vector expression would be an efficient computation method. In particular, the spatio-temporal image sequence of significance is the result of dynamic and static characteristics of integrated stimulus.

Spatio-temporal saliency perception is a rapid processing result of multi-features contrasting in parallel in multi-scales.

The position of a pixel is important to saliency detection in an image, since people tend to focus their attention on some specific areas.

The remainder of this paper is organized as follows: in Section 2, we summarize and analyze existing algorithms. Section 3 gives the details of our visual saliency perception model, including spatio-temporal hypercomplex spectral contrast computation, a log-polar bias sampling strategy and saliency map computation. Section 4 presents and discusses the experimental results and evaluations for our model by comparing the proposed approach with other state-of-the-art methods on more than 1,000 natural and psychological images. In Section 5, we discuss the difference between and the proposed algorithm with other related methods. Section 6 explores the application of the proposed approach in moving object extraction in dynamic scenes. Finally, conclusions and future works are given in Section 7.

## 3. Our Approach

In this section, we describe the proposed model in detail. The framework of our approach is illustrated in Figure 2. In this work, we compute a hypercomplex Fourier spectrum contrast of the amplitude and phase information using hypercomplex Fourier transform, respectively, in the multi-scale HSV color space. In this case, the saliency map could be produced using two hypercomplex spectral contrast maps at the same time by reconstruction and non-uniform sampling. The proposed HSC method mainly contains four steps:

**Step 1:**Convert a raw image,**I**, to the HSV color space, and then**I**was blurred by 2D Gaussian on three level pyramids to eliminate fine texture details, as well as to average the energy of image**I**.**Step 2:**Represent image pixels by pure quaternion (hypercomplex) on HSV color space, then calculate the hypercomplex Fourier spectrum, which contains amplitude and phase information of the image by hypercomplex Fourier transform [41] in different scales.**Step 3:**Calculate the spectral contrast between the raw image and blurred image, and then, reconstruct these contrast maps using amplitude spectral and phase spectral under various scales of the raw image.**Step 4:**Normalize the reconstructed spectral contrast maps and use log-polar non-uniform sampling to obtain the final saliency map.

#### 3.1. Hypercomplex of HSV Color Image

Quaternion is a kind of hypercomplex number. Color image pixels have inherently 3-D components, and they can be represented in quaternion form using pure quaternion [41]. A commonly used color space that corresponds more naturally to human perception is the HSV color space, which contains three components: hue, saturation and value. In this paper, each pixel of the raw image is represented by hypercomplex numbers (quaternion) consisting of HSV three-color components, which do not consider color opponent-component (RG or BY) and intensity, different from [27,40]. Thus, a hypercomplex number HSV image q(x,y) is defined as follows:

where**i,j,k**satisfies

**i**

^{2}=

**j**

^{2}=

**k**

^{2}= −1,

**i**⊥

**j**,

**j**⊥

**k**,

**i**⊥

**k**,

**k**=

**ij**.

Based onp the definition above, the hypercomplex number HSV image q's pixel is given by pixel symplectic decomposition as:

#### 3.2. Saliency Detection Using HSC

Usually, salient visual stimulus is often generated by strong contrast signals in the bottom-up model, which have a larger energy of spectrum. In another words, some strong spectral contrast of amplitude and phase are the main components in salient signals. In this paper, we calculate the amplitude spectrum and phase spectrum using the hypercomplex Fourier transform [41] of the HSV color image. Based on Equation (2), hypercomplex Fourier transform of the hypercomplex image, q, can be calculated by two complex Fourier transforms of the symplectic parts, such as:

We define each part of the forward and inverse hypercomplex Fourier Transform of Equation (3) in Equation (4):

In mathematics, **hypercomplex analysis** is the extension of real analysis and complex analysis to the study of functions where the argument is a hypercomplex number. The first instance is functions of a quaternion variable, where the argument is a quaternion. A second instance involves functions of a motor variable where arguments are split-complex numbers.

In mathematical physics, there are hypercomplex systems called Clifford algebras. The study of functions with arguments from a Clifford algebra is called Clifford analysis.

A matrix may be considered a hypercomplex number. For example, study of functions of 2 × 2 real matrices shows that the topology of the space of hypercomplex numbers determines the function theory. Functions such as square root of a matrix, matrix exponential, and logarithm of a matrix are basic examples of hypercomplex analysis.^{[1]} The function theory of diagonalizable matrices is particularly transparent since they have eigendecompositions.^{[2]} Suppose where the E_{i} are projections. Then for any polynomial

Modern terminology is *algebra* for "system of hypercomplex numbers", and the algebras used in applications are often Banach algebras since Cauchy sequences can be taken to be convergent. Then the function theory is enriched by sequences and series. In this context the extension of holomorphic functions of a complex variable is developed as the holomorphic functional calculus. Hypercomplex analysis on Banach algebras is called functional analysis.

## See also[edit]

## References[edit]

- Daniel Alpay (editor) (2006)
*Wavelets, Multiscale systems and Hypercomplex Analysis*, Springer, ISBN 9783764375881 . - Enrique Ramirez de Arellanon (1998)
*Operator theory for complex and hypercomplex analysis*, American Mathematical Society (Conference proceedings from a meeting in Mexico City in December 1994). - Geoffrey Fox (1949)
*Elementary Function Theory of a Hypercomplex Variable and the Theory of Conformal Mapping in the Hyperbolic Plane*, M.A. thesis, University of British Columbia. - Sorin D. Gal (2004)
*Introduction to the Geometric Function theory of Hypercomplex variables*, Nova Science Publishers, ISBN 1-59033-398-5. - R. Lavika & A.G. O’ Farrell & I. Short(2007) "Reversible maps in the group of quaternionic Möbius transformations", Mathematical Proceedings of the Cambridge Philosophical Society 143:57–69.
- Birkhauser Mathematics (2011)
*Hypercomplex Analysis and Applications*, series with editors Irene Sabadini and Franciscus Sommen. - Irene Sabadini & Michael V. Shapiro & F. Sommen (editors) (2009)
*Hypercomplex Analysis*, Birkhauser ISBN 978-3-7643-9892-7. - Springer (2012)
*Advances in Hypercomplex Analysis*, eds Sabadini, Sommen, Struppa.

## External links[edit]

**^**Felix Gantmacher (1959)*The Theory of Matrices*, two volumes, translator: Kurt Hirsch, Chelsea Publishing, chapter 5: functions of matrices, chapter 8: roots and logarithms of matrices**^**Shaw, Ronald (1982)*Linear Algebra and Group Representations*, v. 1, § 2.3, Diagonalizable linear operators, pages 78–81, Academic PressISBN 0-12-639201-3.

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